Problem: Let  \[A=111111\]and  \[B=142857\]Find a positive integer $N$ with six or fewer digits such that $N$ is the multiplicative inverse of $AB$ modulo 1,000,000.
We notice that both $A$ and $B$ are factors of 999,999.  Specifically \[9A=999999\]and \[7B=999999.\]Taken modulo 1,000,000 these equations read \begin{align*}
9A&\equiv-1\pmod{1{,}000{,}000}\\
7B&\equiv-1\pmod{1{,}000{,}000}\\
\end{align*}We are set if we multiply these equations: \[(9A)(7B)\equiv1\pmod{1{,}000{,}000}\]so $N=9\cdot7=\boxed{63}$ is the multiplicative inverse to $AB$ modulo 1,000,000.